Convergence of the iterated Aluthge transform sequence for diagonalizable matrices

Given an r x r complex matrix T, if T = U vertical bar T vertical bar is the polar decomposition of T, then, the Aluthge transform is defined by Delta(T) = vertical bar T vertical bar U-1/2 vertical bar T vertical bar(1/2). Let Delta(n)(T) denote the n-times iterated Aluthge transform of T, i.e. Delta(0)(T) = T and Delta(n)(T) = Delta(n)(T) = Delta(Delta(n-1)(T)), n is an element of N. We prove that the sequence [Delta(n)(T))(n is an element of N) converges for every r x r diagonalizable matrix T. We show that the limit Delta(infinity)(.) is a map of class C(infinity)on the similarity orbit of a diagonalizable matrix, and on the (open and dense) set of r x r matrices with r different eigenvalues. (c) 2007 Elsevier Inc. All rights reserved.